Hint. If you bring $\def\R{\Bbb R}\R^n$ in bijection with a hyperplane $H$ in $\R^{n+1}$ that does not pass through the origin (for instance by adding a final coordinate $1$ to vectors), then every translation by a vector in $\R^n$ corresponds to a translation $H\to H$ that can be uniquely extended to a linear operator $\R^{n+1}\to\R^{n+1}$ (which of course must stabilise globally the hyperplane $H$).